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<H3>fix ttm command 
</H3>
<P><B>Syntax:</B>
</P>
<PRE>fix ID group-ID ttm seed C_e rho_e kappa_e gamma_p gamma_s v_0 Nx Ny Nz T_infile N T_outfile 
</PRE>
<UL><LI>ID, group-ID are documented in <A HREF = "fix.html">fix</A> command
<LI>ttm = style name of this fix command
<LI>seed = random number seed to use for white noise (positive integer)
<LI>C_e  = electronic specific heat (energy/(electron*temperature) units)
<LI>rho_e = electronic density (electrons/volume units)
<LI>kappa_e = electronic thermal conductivity (energy/(time*distance*temperature) units)
<LI>gamma_p = friction coefficient due to electron-ion interactions (mass/time units)
<LI>gamma_s = friction coefficient due to electronic stopping (mass/time units)
<LI>v_0 = electronic stopping critical velocity (velocity units)
<LI>Nx = number of thermal solve grid points in the x-direction (positive integer)
<LI>Ny = number of thermal solve grid points in the y-direction (positive integer)
<LI>Nz = number of thermal solve grid points in the z-direction (positive integer)
<LI>T_infile = filename to read initial electronic temperature from
<LI>N = dump TTM temperatures every this many timesteps, 0 = no dump
<LI>T_outfile = filename to write TTM temperatures to (only needed if N > 0) 
</UL>
<P><B>Examples:</B>
</P>
<PRE>fix 2 all ttm 699489 1.0 1.0 10 0.1 0.0 2.0 1 12 1 initialTs 1000 T.out
fix 2 all ttm 123456 1.0 1.0 1.0 1.0 1.0 5.0 5 5 5 Te.in 1 Te.out 
</PRE>
<P><B>Description:</B>
</P>
<P>Use a two-temperature model (TTM) to represent heat transfer through
and between electronic and atomic subsystems.  LAMMPS models the
atomic subsystem as usual with a molecular dynamics model and the
classical force field specified by the user, but the electronic
subsystem is modeled as a continuum, or a background "gas", on a
regular grid.  Energy can be transferred spatially within the grid
representing the electrons.  Energy can also be transferred between
the electronic and the atomic subsystems.  The algorithm underlying
this fix was derived by D. M.  Duffy and A. M. Rutherford and is
discussed in two J Physics: Condensed Matter papers: <A HREF = "#Duffy">(Duffy)</A>
and <A HREF = "#Rutherford">(Rutherford)</A>.  They used this algorithm in cascade
simulations where a primary knock-on atom (PKA) was initialized with a
high velocity to simulate a radiation event.
</P>
<P>Heat transfer between the electronic and atomic subsystems is carried
out via an inhomogeneous Langevin thermostat.  This thermostat differs
from the regular Langevin thermostat (<A HREF = "fix_langevin.html">fix
langevin</A>) in three important ways.  First, the
Langevin thermostat is applied uniformly to all atoms in the
user-specified group for a single target temperature, whereas the TTM
fix applies Langevin thermostatting locally to atoms within the
volumes represented by the user-specified grid points with a target
temperature specific to that grid point.  Second, the Langevin
thermostat couples the temperature of the atoms to an infinite heat
reservoir, whereas the heat reservoir for fix TTM is finite and
represents the local electrons.  Third, the TTM fix allows users to
specify not just one friction coefficient, but rather two independent
friction coefficients: one for the electron-ion interactions
(<I>gamma_p</I>), and one for electron stopping (<I>gamma_s</I>).
</P>
<P>When the friction coefficient due to electron stopping, <I>gamma_s</I>, is
non-zero, electron stopping effects are included for atoms moving
faster than the electron stopping critical velocity, <I>v_0</I>.  For
further details about this algorithm, see <A HREF = "#Duffy">(Duffy)</A> and
<A HREF = "#Rutherford">(Rutherford)</A>.
</P>
<P>Energy transport within the electronic subsystem is solved according
to the heat diffusion equation with added source terms for heat
transfer between the subsystems:
</P>
<CENTER><IMG SRC = "Eqs/fix_ttm.jpg">
</CENTER>
<P>where C_e is the specific heat, rho_e is the density, kappa_e is the
thermal conductivity, T is temperature, the "e" and "a" subscripts
represent electronic and atomic subsystems respectively, g_p is the
coupling constant for the electron-ion interaction, and g_s is the
electron stopping coupling parameter.  C_e, rho_e, and kappa_e are
specified as parameters to the fix.  The other quantities are derived.
The form of the heat diffusion equation used here is almost the same
as that in equation 6 of <A HREF = "#Duffy">(Duffy)</A>, with the exception that the
electronic density is explicitly reprensented, rather than being part
of the the specific heat parameter.
</P>
<P>Currently, this fix assumes that none of the user-supplied parameters
will vary with temperature.  This assumption can be relaxed by
modifying the source code to include the desired temperature
dependency and functional form for any of the parameters.  Note that
<A HREF = "#Duffy">(Duffy)</A> used a tanh() functional form for the temperature
dependence of the electronic specific heat, but ignored temperature
dependencies of any of the other parameters.
</P>
<P>This fix requires use of periodic boundary conditions and a 3D
simulation.  Periodic boundary conditions are also used in the heat
equation solve for the electronic subsystem.  This varies from the
approach of <A HREF = "#Rutherford">(Rutherford)</A> where the atomic subsystem was
embedded within a larger continuum representation of the electronic
subsystem.
</P>
<P>The initial electronic temperature input file, <I>T_infile</I>, is a text
file LAMMPS reads in with no header and with four numeric columns
(ix,iy,iz,Temp) and with a number of rows equal to the number of
user-specified grid points (Nx by Ny by Nz).  The ix,iy,iz are node
indices from 0 to nxnodes-1, etc.  For example, the initial electronic
temperatures on a 1 by 2 by 3 grid could be specified in a <I>T_infile</I>
as follows:
</P>
<PRE>0 0 0 1.0
0 0 1 1.0
0 0 2 1.0
0 1 0 2.0
0 1 1 2.0
0 1 2 2.0 
</PRE>
<P>where the electronic temperatures along the y=0 plane have been set to
1.0, and the electronic temperatures along the y=1 plane have been set
to 2.0.  The order of lines in this file is no important.  If all the
nodal values are not specified, LAMMPS will generate an error.
</P>
<P>The temperature output file, <I>T_oufile</I>, is created and written by
this fix.  Temperatures for both the electronic and atomic subsystems
at every node and every N timesteps are output.  If N is specified as
zero, no output is generated, and no output filename is needed.  The
format of the output is as follows.  One long line is written every
output timestep.  The timestep itself is given in the first column.
The next Nx*Ny*Nz columns contain the temperatures for the atomic
subsystem, and the final Nx*Ny*Nz columns contain the temperatures for
the electronic subsystem.  The ordering of the Nx*Ny*Nz columns is
with the z index varing fastest, y the next fastest, and x the
slowest.
</P>
<P>This fix does not change the coordinates of its atoms; it only scales
their velocities.  Thus a time integration fix (e.g. <A HREF = "fix_nve.html">fix
nve</A>) should still be used to time integrate the affected
atoms.  This fix should not normally be used on atoms that have their
temperature controlled by another fix - e.g. <A HREF = "fix_nh.html">fix nvt</A> or
<A HREF = "fix_langevin.html">fix langevin</A>.
</P>
<P>This fix computes 2 output quantities stored in a vector of length 2,
which can be accessed by various <A HREF = "Section_howto.html#howto_15">output
commands</A>.  The first quantity is the
total energy of the electronic subsystem. The second quantity is the
energy transferred from the electronic to the atomic subsystem on that
timestep. Note that the velocity verlet integrator applies the fix ttm
forces to the atomic subsystem as two half-step velocity updates: one
on the current timestep and one on the subsequent timestep.
Consequently, the change in the atomic subsystem energy is lagged by
half a timestep relative to the change in the electronic subsystem
energy. As a result of this, users may notice slight fluctuations in
the sum of the atomic and electronic subsystem energies reported at
the end of the timestep.
</P>
<P>The vector values calculated by this fix are "extensive".
</P>
<P>IMPORTANT NOTE: The current implementation creates a copy of the
electron grid that overlays the entire simulation domain, for each
processor.  Values on the grid are summed across all processors.  Thus
you should insure that this grid is not too large, else your
simulation could incur high memory and communication costs.
</P>
<P><B>Restart, fix_modify, output, run start/stop, minimize info:</B>
</P>
<P>This fix writes the state of the electronic subsystem and the energy
exchange between the subsystems to <A HREF = "restart.html">binary restart
files</A>.  See the <A HREF = "read_restart.html">read_restart</A> command
for info on how to re-specify a fix in an input script that reads a
restart file, so that the operation of the fix continues in an
uninterrupted fashion.
</P>
<P>Because the state of the random number generator is not saved in the
restart files, this means you cannot do "exact" restarts with this
fix, where the simulation continues on the same as if no restart had
taken place.  However, in a statistical sense, a restarted simulation
should produce the same behavior.
</P>
<P>None of the <A HREF = "fix_modify.html">fix_modify</A> options are relevant to this
fix.  No global or per-atom quantities are stored by this fix for
access by various <A HREF = "Section_howto.html#howto_15">output commands</A>.  No
parameter of this fix can be used with the <I>start/stop</I> keywords of
the <A HREF = "run.html">run</A> command.  This fix is not invoked during <A HREF = "minimize.html">energy
minimization</A>.
</P>
<P><B>Restrictions:</B>
</P>
<P>This fix can only be used for 3d simulations and orthogonal
simlulation boxes.  You must use periodic <A HREF = "boundary.html">boundary</A>
conditions with this fix.
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "fix_langevin.html">fix langevin</A>, <A HREF = "fix_dt_reset.html">fix dt/reset</A>
</P>
<P><B>Default:</B> none
</P>
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<A NAME = "Duffy"></A>

<P><B>(Duffy)</B> D M Duffy and A M Rutherford, J. Phys.: Condens. Matter, 19,
016207-016218 (2007).
</P>
<A NAME = "Rutherford"></A>

<P><B>(Rutherford)</B> A M Rutherford and D M Duffy, J. Phys.:
Condens. Matter, 19, 496201-496210 (2007).
</P>
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